On regular rings whose maximal right quotient rings are type I_f

Abstract

This paper is written about the property (DF) on regular rings whose maximal right quotient rings are Type If. Hereafter regular rings whose maximal right quotient rings are Type If are said to satisfy (*). The property (DF) is very important property when we study on regular rings satisfying (*), and it was treated in the paper [5] written by the first author, where (DF) for a ring R is defined as that if the direct sum of any two directly finite projective /^-modules is always directly finite. In the above paper, the equivalent condition that a regular ring R of bounded index satisfies (DF) was discovered and called (#). Stillmore, we proved that the condition (DF) is equivalent to (#) for regular rings whose primitive factor rings are artinian in the paper [6]. Then we have the problem that (DF) is equivalent (#) for regular rings satisfying (*) or not, where the condition (*) is weaker than one that primitive factor rings are artinian. In § 2, we shall prove Theorem 2.4. This is important, and using this, Theorem 2.5 (i.e. if R is a regular ring satisfying (*) and k is any positive integer, then kP is directly finite for every directly finite projective Tΐ-module P) is proved. Moreover, we shall solve the above problem in Theorem 2.11. In § 3, we shall consider some applications of Theorem 2.11. We prove Theorem 3.3 that if R is a regular ring satisfying (*) whose maximal right quotient ring of R satisfies (DF), then so does R. Though it is clear that a regular rings satisfying (*) which has a nonzero essential socle satisfies (DF), we can prove that, for regular rings satisfying (*), the condition having a nonzero essential socle is not equivalent to (») in Example 3.4. Next, we shall consider that (ΠfR)/(®R) satisfies (DF) or not for a regular ring R satisfying (*). This problem is a generalization of Example 3.4, and we prove that, for a regular ring R of bounded index, (TL?R)/(®R) satisfies (DF) (Theorem 3.9).